"Operations"
is mathematician-ese for "procedures". The four "basic
operations" on numbers are addition, subtraction, multiplication,
and division. For matrices, there are three basic row operations; that
is, there are three procedures that you can do with the rows of a matrix.

The first operation is
row-switching. For instance, given the matrix:

...you can switch the rows
around to put the matrix into a nicer row arrangement, like this:

Row-switching is often
indicated by drawing arrows, like this:

When switching rows around,
be careful to copy the entries correctly.

The second operation is
row multiplication. For instance, given the following matrix:

...you can multiply the
first row by–1to get a positive leading
value in the first row:

This row multiplication
is often indicated by using an arrow with multiplication listed on top
of it, like this:

The "–1R1"
indicates the actual operation. The "–1"
says that we multiplied by negative one; the "R1"
says that we were working with the first row. Note that the second and
third rows were copied down, unchanged, into the second matrix. The multiplication
only applied to the first row, so the entries for the other two rows were
just carried along unchanged.

You can multiply by anything
you like. For instance, to get a leading1
in the third row of the previous matrix, you can multiply the third row
by a negative one-half:

Since you weren't doing
anything with the first and second rows, those entries were just copied
over unchanged into the new matrix.

You can do more than one
row multiplication within the same step, so you could have done the two
above steps in just one step, like this:

It is a good idea to use
some form of notation (such as the arrows and subscripts above) so you
can keep track of your work. Matrices are very messy, especially if you're
doing them by hand, and notes can make it easier to check your work later.
It'll also impress your teacher.

The last row operation
is row addition. Row addition is similar to the "addition" method
for solving
systems of linear equations.
Suppose you have the following system of equations:

You could start solving
this system by adding down the columns to get4y
= 4:

You can do something similar
with matrices.For instance, given the following matrix:

...you can "reduce"
(get more leading zeroes in) the second row by adding the first row to
it (the general goal with matrices at this stage being to get a "1"
— or "0's"
and then a "1"
— at the beginning of each matrix row). When you were reducing the two-equation
linear system by adding, you drew an "equals" bar across the
bottom and added down. When you are using addition on a matrix, you'll
need to grab some scratch paper, because you don't want to try to do the
work inside the matrix. So add the two rows on your scratch paper:

Scratch work —
don't hand this in!

This is your new second
row; you will write it in place of the old second row. The result will
look like this:

In this case, the "R1
+ R2"
on the arrow means "I added row one to row two, and this is the result
I got". Since row one didn't actually change, and since we didn't
do anything with row three, these rows get copied into the new matrix
unchanged.